Electric Waves through Tubes. 129 



All the values of k determined by (2) and (7), or by (21) 

 and (22) , are real, but the reality of k still leaves it open 

 whether m in (3) shall be real or imaginary. If we are 

 dealing with free stationary vibrations m is given and real, 

 from which it follows that p is also real. But if it hep that is 

 given, m 2 may be either positive or negative, In the former 

 case the motion is really periodic with respect to z ; but in the 

 latter z enters in the forms e m ' z , e~ m ' z , and the motion becomes 

 infinite when z=+co,or when z=— go , or in both cases. 

 If the smallest of the possible values of k 2 exceeds p' 2 /V 2 , in is 

 necessarily imaginary, that is to say no periodic waves of the 

 frequency in question can be propagated along the cylinder. 



Rectangular Section. 



The simplest case to which these formula? can be applied is 

 when the section of the cylinder is rectangular, bounded, we 

 may suppose, by the lines # = 0, x = a, y = 0, y = /3. 



As for the vibrations of stretched membranes,* the appro- 

 priate value of R applicable to solutions of the first class is 



R- e i{mz+ P t) sin (pnxj^ s i n ^1$) ; . . . (23) 



from which the remaining functions are deduced so easily 

 by (15), (16) that it is hardly necessary to write down the 

 expressions. In (23) yu and v are integers, and by (13) 



^6+0 •••• ;■<") 



m*=f/V*-Tr>(£+ v l). . . . (25) 



The lowest frequency which allows of the propagation of 

 periodic waves along the cylinder is given by 



^ __ 7T 2 7T 2 



y2 — 2 "*" ~o2 (26) 



If the actual frequency of a vibration having its origin at any 

 part of the cylinder be much less than the above, the resulting 

 disturbance is practically limited to a neighbouring finite 

 length of the cylinder. 



For vibrations of the second class we have 



c = e^ nz+ ^cos(/i7rx/cc)cos{v7rg//3), . . . (27) 



the remaining functions being at once deducible by means of 

 (19), (20). The satisfaction of (22) requires that here again 



* • Theory of Sound/ § 195. 



whence 



